Generalized D-symmetric operators. II. (English) Zbl 1251.47032
Summary: Let \(H\) be a separable, infinite-dimensional, complex Hilbert space and let \(A, B\in\mathcal L(H)\), where \(\mathcal L(H)\) is the algebra of all bounded linear operators on \(H\). Let \(\delta_{AB}: \mathcal L(H)\rightarrow \mathcal L(H)\) denote the generalized derivation \(\delta_{AB}(X)=AX-XB\). This note will initiate a study on the class of pairs \((A,B)\) such that \(\overline{\mathcal R(\delta_{AB})}= \overline{\mathcal R(\delta_{A^{\ast}B^{\ast}})}\).
For Part I, see [Serdica Math. J. 34, No. 3, 557–562 (2008; Zbl 1216.47062)].
For Part I, see [Serdica Math. J. 34, No. 3, 557–562 (2008; Zbl 1216.47062)].
MSC:
| 47B47 | Commutators, derivations, elementary operators, etc. |
| 47B10 | Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) |
| 47A30 | Norms (inequalities, more than one norm, etc.) of linear operators |
